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Relation between window boundary dimension and diffraction decay

Determine the relationship between the Hausdorff dimension of the boundary of the window in a cut-and-project (model set) description and the asymptotic decay rate of the diffraction measure’s intensities, thereby clarifying whether and how boundary dimension governs diffraction decay for such model sets.

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Background

In Section 2.4, the paper compares two one-dimensional Pisot substitution examples whose model-set windows have markedly different boundary structures: one with interval windows (smooth boundary) and another with Rauzy-fractal windows (Cantorval boundary of non-integer Hausdorff dimension).

The authors compute and compare diffraction intensities and observe a slower decay in the case with fractal-boundary windows. They note that this observation supports a conjecture that the decay of diffraction is influenced by the boundary dimension of the window, citing earlier work.

The precise dependence between boundary dimension and diffraction decay, however, is not established in the paper, indicating an open conjectural relationship.

References

On the level of intensities, one can recognise that the decay of I_{\widetilde{\varrho}}(k) is slower than that of I{}_{\varrho}(k), which supports the conjectured non-trivial relation between the boundary dimension of the window and the decay rate of the diffraction measure.

— Diffraction of the Hat and Spectre tilings and some of their relatives (2502.03268 - Baake et al., 5 Feb 2025) in Section 2.4 (Some variations on the guiding example)